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31 October, 03:22

There are 14 juniors and 16 seniors in a chess club. a) From the 30 members, how many ways are there to arrange 5 members of the club in a line? b) How many ways are there to arrange 5 members of the club in a line if there must be a senior at the beginning of the line and at the end of the line? 0 c) If the club sends 2 juniors and 2 seniors to the tournament, how many possible groupings are there? d) If the club sends either 4 juniors or 4 seniors, how many possible groupings are there?

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  1. 31 October, 06:40
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    a) 17,100,720

    b) 4,717,440

    c) 10,920

    d) 2821

    Step-by-step explanation:

    14 juniors and 16 seniors = 30 people

    a) From the 30 members, how many ways are there to arrange 5 members of the club in a line?

    As it is a ordered arrangement

    30.29.28.27.26 = 17,100,720

    b) How many ways are there to arrange 5 members of the club in a line if there must be a senior at the beginning of the line and at the end of the line?

    16.28.27.26.15 = 4,717,440

    c) If the club sends 2 juniors and 2 seniors to the tournament, how many possible groupings are there?

    Not ordered arrangement. And means that we need to multiply the results.

    C₁₄,₂ * C₁₆,₂

    C₁₄,₂ = 14.13.12! = 14.13 = 91

    12! 2! 2

    C₁₆,₂ = 16.15.14! = 16.15 = 120

    14! 2! 2

    C₁₄,₂ * C₁₆,₂ = 91.120 = 10,920

    d) If the club sends either 4 juniors or 4 seniors, how many possible groupings are there?

    Or means that we need to sum the results.

    C₁₄,₄ + C₁₆,₄

    C₁₄,₄ = 14.13.12.11.10! = 14.13.12.11 = 1001

    10! 4! 4.3.2.1

    C₁₆,₄ = 16.15.14.13.12! = 16.15.14.13 = 1820

    12! 4! 4.3.2.1

    C₁₄,₄ + C₁₆,₄ = 1001 + 1820 = 2821
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